Human language

In this main section I show how human language generates necessary and impossible sentences, and how we are able to recognize them. I shall understand by a ‘factual sentence’ one which is recognized in a language-speaking community as true under a certain range of conditions and false under a certain non-overlapping range of conditions. We learn which sentences are factual by hearing many examples of them which are treated by speakers and hearers in a certain kind of way. These factual sentences are of various kinds classified by logicians – subject-predicate, existential generalization, hypothetical etc. Factual sentences are contrasted with sentences having other roles, e.g. performative utterances or questions. Any sentence of any of the same forms as a factual sentence I shall call a ‘declarative’ sentence.

We learn the meanings of different words by seeing the contributions which they make to sentences which include them. More specifically, we learn the meaning of a word by being taught the observable conditions under which sentences in which that word plays a certain role are regarded as true or false, and by the other sentences to which a speaker is regarded as committed by uttering some sentence containing that word in a particular role, and the other sentences which are such that someone who utters them is regarded as committed to a sentence containing that word in a particular role. We learn the meaning of ‘lion’ for example by seeing the circumstances in which ‘there is a lion over there’ is regarded as true, and the circumstances in which it is regarded as false; and by the fact that someone who utters this sentence is regarded as committed to ‘there is an animal over there’, and someone who utters ‘there are two lions over there’ is regarded as committed to the original sentence.

We need to observe different paradigm examples of the observable conditions under which a sentence containing a certain word in various roles is regarded as true or false, and of the commitments speakers who use sentences containing that word in various roles are regarded as having; and this allows us to understand the conditions under which some new sentence containing that word would be regarded as true or false. We extrapolate, that is, from a stock of past examples (of observable conditions and relations of commitment) to an understanding that the sentence would be regarded as true (or false, as the case may be) under conditions sufficiently similar to the paradigm examples.

This learning process gives rise to an indeterminacy or vagueness in the truth- and falsity-conditions of sentences, and so in the public understanding of the meaning of any sentence in three distinct ways. First, there may be borderline cases of conditions ‘sufficiently similar to the paradigm examples’ of the truth-conditions. If someone is taught the truth- and falsity-conditions of ‘This is blue’ by certain examples of blue things contrasted with certain examples of non-blue things (e.g. green things or red things), he will not know whether to say of a certain bluish-green thing whether it is blue or whether it is not blue. Secondly, two speakers taught the meaning of a sentence by the same paradigm examples may have slightly different psychological make-ups, as a result of which some of them extrapolate differently from the paradigm examples so as to make different judgments about new examples. And thirdly, different speakers learn the truth- and falsity-conditions of a sentence by being shown slightly different examples of observable conditions under which the sentence is regarded as true or false, and by being given different examples of sentences to which a speaker is regarded as committed by the original sentence and which are such that someone who asserts them is regarded as committed to the original sentence.

Indeterminacy limits interpersonal communication and reasoning; and so dictionary compilers, grammarians, logicians, and other philosophers have sought to lay down rules for correct usage, usually by codifying most people's actual usage. The rules seek to summarize in a few sentences the observable conditions under which various sentences of the language are true and under which observable conditions they are false, and the relations of commitment which sentences have to other sentences. Given a ‘correct’ use, a sentence then has a public objective meaning, and the distinction emerges between the observable conditions which a particular speaker regards as making a particular sentence true (or false, as the case may be) and the conditions which do in fact make that sentence true (or false); and between the commitments in respect of other sentences which a particular speaker regards himself as having by uttering a particular sentence and those he actually (objectively) has by uttering that sentence.

Sometimes ‘tidying up’ leads to recognition of two distinct senses of an old word. Sometimes it leads to elimination of a minority usage. But the brief rule-stating sentences need to be given content by examples, and insofar as people learn meanings from different paradigm examples for the application and commitments of sentences, and extrapolate from them differently, the indeterminacy will remain. Of course each speaker will believe that he is using words in a way which his hearers will understand, and so that he is conforming to any correct usage; but he may not be; and even if he is, that usage may be such that a sentence may be uttered under conditions which lie between its truth-conditions and its falsity-conditions.

Which sentences remain vague, and the degree of their vagueness, depends on how language is tidied up. But while it is, I suggest, implausible to suppose that it could be tidied up in such a way that no vagueness remains (as the epistemic theory claims to be the actual situation), or so that vagueness is replaced by ‘degrees of truth’, codification of language in such a way that ‘supervaluation’ or ‘gap’ semantics applies is clearly possible, and I cannot understand why Leftow claims (144) that my views rule that out. In such a public language Leftow's (144) DC: ‘If the rules commit one to P, it is determinate that they commit him/her to P’, will nevertheless hold; and I am puzzled as to why Leftow should think that this constitutes a problem for my view. It is desirable for a public language that there be a fairly precise understanding of the notion of negation; as I shall understand it, the negation of a sentence s is a sentence which has the same truth-conditions as s has falsity-conditions, and conversely. In philosophical codification of English, ‘~s’ is often used to designate any device which has this role of negating s.

We may call a rule for what one is objectively committed to by a sentence, a rule of mini-entailment. s ₁ mini-entails s ₂ if and only if someone who asserts s ₁ is thereby (in virtue of the rules for the correct use of language) committed to s ₂. s ₁ entails s _n if they are joined by a chain of mini-entailments, such that s ₁ mini-entails some s ₂, s ₂ mini-entails some s ₃ and so on until we reach a sentence which mini-entails s _n. Not all speakers will have been taught the same mini-entailments of a given sentence; but as long as they have been taught many of the same mini-entailments and have a similar psychological make-up, they may naturally come to assume that the other mini-entailments taught to others also hold. Or, using some mini-entailments, they may be able to derive as derivative entailments what others have been taught as mini-entailments involved in the meaning of the sentence. And if people generally recognize some entailment, it becomes a mini-entailment. If this process of coming to agree about entailments happens fairly quickly and naturally, it becomes natural to say that speakers are just sorting out what is involved in their common concept. They are ensuring that they are using words and sentence-forms in the same more precise senses as each other, but not thereby depriving themselves of a useful sense.

Given a standard ‘correct’ language, it then becomes apparent that words can be combined into declarative sentences (i.e. sentences of one of the same forms as paradigm factual sentences) in such a way that the rules of the language alone suffice to ensure that they cannot be true; and that words can be combined into declarative sentences in such a way that the rules of the language alone suffice to ensure that their negations cannot be true. The former sentences (e.g. ‘All squares are round’) are the logically impossible sentences, the latter (e.g. ‘No squares are round’) the logically necessary sentences; they are false or true (as the case may be) ‘solely in virtue of the meanings of the words which they contain’ and, I must add, ‘the way those words are arranged’. There are no circumstances under which the former could be true, and so – it is natural to say – they have no truth-conditions (merely falsity-conditions). There are no circumstances under which the latter could be false, and so – it is natural to say – they have no falsity-conditions.Footnote ⁸ (The ‘no circumstances under which they could be’ true or false is merely a matter of it following from the rules of language, not a matter of a necessity independent of language.)

The paradigm example of a sentence which has no truth-conditions is a self-contradictory sentence, which is a sentence which claims both that something is so and also that it is not so (normally expressed by a sentence of the form ‘s and not-s’). For such a sentence could only be true if that something was so, and the sentence asserts that it is not so. But it is not, and (despite Leftow's claim in various places) I hope that I have never written in print that it is,Footnote ⁹ a conventional matter that PNC (the principle of non-contradiction) is true. What arises from linguistic convention is that there are declarative sentences which assert both that something is so and also that that same something is not so, and which sentences they are. Any sentence entailed by a logically necessary sentence will also be logically necessary; for since the former could not be false, neither could any sentence to which (in virtue of the rules of the language) speakers are committed to by it be false. Likewise any sentence which entails a logically impossible sentence will also be logically impossible since if (in virtue of the rules of the language) speakers are committed by some sentence to some other sentence which could not be true, the former sentence itself could not be true.

Logical necessity and possibility are often equated with conceivability and inconceivability.Footnote ¹⁰ Logically possible sentences are such that it is conceivable that they are true – which we may abbreviate as ‘they are conceivable’; logically necessary sentences are such that it is inconceivable that their negations are true – which we may abbreviate as ‘their negations are inconceivable’. ‘Conceivable’ may be used in either an objective or a subjective sense. In the objective sense, a sentence is conceivable iff some speaker of the language could understand what it would be like for that sentence to be true, that is could know what are the truth conditions of that sentence. ‘Understanding what it would be like’ for a sentence to be true, or ‘knowing the truth-conditions’ of a sentence are matters of degree, but – whatever the degree involved – both entail the sentence having truth-conditions. And since the rules of a language and their consequences are knowable (at least to a considerable extent), it follows that a sentence will be conceivable (in the objective sense) if and only if the rules of the language have the consequence that it has truth-conditions. So the objectively conceivable is the logically possible; the inconceivable is the logically impossible; that, the negation of which is inconceivable, is the logically necessary.

In the subjective sense, a sentence is conceivable if some speaker of the language could think that she understood what it would be like to be true in the sense of thinking that she knows what its truth-conditions are. Subjective conceivability does not guarantee objective conceivability, but it is evidence of it. If it seems to you that you understand what the truth-conditions of a sentence are, that is evidence that it does have truth-conditions (and the evidence is stronger, the more strongly it seems to you that you do understand this). That follows from the very general principle of credulity that (for any p) the more strongly it seems to you that p, the more probable it is – on your evidence – that p. In due course I shall add to this general principle an additional strong reason for supposing that it applies in this case, that the subjectively conceivable is probably objectively conceivable. Henceforward I shall use ‘conceivable’ in the objective sense.

Leftow's (148) claim that ‘once it is determinate what [sentences] say, whether we can conceive what they say is a fact about our powers of conception’ is false, if it is read as implying that with different powers of conception we could conceive a given sentence (with a correct meaning in the language) which was previously inconceivable. For what sentences say determines whether or not they are conceivable. Another way which I have used of describing the inconceivability of a declarative sentence is to say that such a sentence lacks ultimate sense; the sentence seems to have a sense, but because of what the rules of language determine is entailed by that sentence, it involves something which it makes no sense to suppose to be true. A declarative sentence which does not involve this then has an ultimate sense.

While I agree with Leftow (154) that there are ‘individual essences we cannot conceive’ (e.g. those of future individuals), no sentence of our language can say anything about such an essence because our language will not have a word which designates that essence. My thesis is a thesis about the inconceivability of sentences of our language, that inconceivability is logically impossibility. In a future language which has a word designating an individual essence currently inconceivable, a sentence about that essence will be conceivable. So Leftow is wrong to draw to draw the momentous conclusion that since ‘the move from inconceivability in our case to impossibility fails’, this ‘undercuts Swinburne's whole modal epistemology’ (154).

A sentence is then in Leftow's terminology ‘narrow-logically’ impossible iff it entails a contradiction and ‘narrow-logically’ necessary iff its negation entails a contradiction;Footnote ¹¹ and that on my account depends on whether there is a sequence of mini-entailments which entail the contradiction. Leftow asserts (141) that I maintain that (in his sense) there are ‘only narrow logical necessities’, that is there are no broad logical necessities. That is not correct. I allow that there are (in my sense) metaphysical necessities discoverable only a posteriori,Footnote ¹² which are a sub-species of what Leftow calls ‘broad logical necessities’. What I do claim, and what Leftow may have in mind, is that all a priori metaphysical necessities are (in Leftow's sense) narrowly logical necessities (that is, such that their negations entail contradictions), and all a priori metaphysical impossibilities are narrowly logical impossibilities (that is, such that they entail contradictions).

To begin with the case of impossibilities – what is asserted could only be a priori impossible if the impossibility is detectable merely by understanding what is involved in what is actually said. To be metaphysically impossible a sentence must have the form of a declarative sentence, in which the component words already have a sense in the language. It will be a subject–predicate sentence, an existential generalization, or some other one of many recognized forms of declarative sentences. It will – to put the point loosely – assert something about some substance or property or event or whatever that it has or does not have some property or relation to some other substance, property etc.; or that there are or are not certain substances, properties, or whatever. Words have a sense insofar as it is clear what are the criteria for an object, property, or whatever being that object, property, or whatever – they therefore normally delimit a boundary to the sort of object or property it can be, or the sort of properties it can have. Hence, it will be inconsistent to affirm that an object picked out by some word is of a kind ruled out by the very criteria for being that object. And the form of a sentence will normally exclude some alternative; and so it will be inconsistent to affirm the sentence together with that alternative. If a sentence is not inconsistent in these ways (or does not entail one that is), we would have no reason to suppose that that sentence is a priori impossible.

It is by its entailing a contradiction (and so being in Leftow's sense ‘narrowly logically impossible’) that we recognize a sentence to be a priori metaphysically impossible, and by its negation entailing a contradiction that we recognize a sentence as a priori metaphysically necessary. Leftow discusses several examples of sentences which seem to be necessary or impossible truths but cannot – he claims – be shown to be so by these criteria. For reasons of space I consider just one such example discussed by Leftow and one discussed by Robert Adams and claim that their accounts of their examples is mistaken. Leftow claims (153), that ‘2=3’, which all can agree is logically impossible, does not entail a contradiction. He acknowledges that ‘2≠3’ follows from the Peano axiom ‘no two natural numbers have the same successor’, and the definitions ‘2=1's successor’, and ‘3=2's successor’; and yet puzzlingly asserts that the joint negation of the axiom and the definitions ‘isn't in form a contradiction and doesn't seem to entail any’ (153). But the meanings of ‘2’ and ‘3’ are defined by the axiom and definitions which Leftow states; and so not merely do the axiom and definitions entail (2≠3); but (2≠3) entails the axiom and definitions. ‘1’ wouldn't designate 1, if it designated a number which could have the same successor as some other number. I don't understand what ‘2≠3’ means unless it is such as to mini-entail and to be mini-entailed by ‘~(2=3)’. So ‘(2=3)’ entails a contradiction (‘(2=3) and ~(2=3)’). There was presumably a time when our number-talk was a lot less precise than it is now and a lot less useful (we've all heard of the tribes whose number system consists of only three words ‘1’, ‘2’, and ‘many’). But now there is a very clear correct usage defined at least by the above Peano axiom and the stated definitions.

Robert Adams discusses the sentence, ‘Everything green has some spatial property’, which he claims ‘seems to be a necessary truth and yet cannot be shown to be “analytic”’.Footnote ¹³ ‘Analytic’ may be understood in different ways, but one way which he mentions is being true ‘solely by virtue of the meanings of its terms’; he claims this account of ‘analytic’ is ‘so vague as to be useless’. I have claimed that, spelled out in terms of the negation entailing a contradiction, the notion is clear and is the notion which I have called ‘narrowly logically necessary’. I suggest that being ‘green’ can be understood in two possible ways, but that the cited sentence with ‘green’ understood in either of these ways can be shown to be logically necessary in my sense. Being green is a property of a thing or substance (in a wide sense). If this thing is a publicly observable thing, then it will exist at a place, and so have a spatial property. Now one might use the word ‘green’ in such a way that being ‘green’ entails being a publicly observable thing: to be green is then to have a certain public characteristic. In that case the negation of the cited sentence clearly entails a contradiction. But there is a sense in which a private thing (accessible only to one conscious being), such as a sense-datum (or, less controversially, an after-image) could be ‘green’. Clearly what it would be for that private thing to be green is to have the same visual appearance in respect of colour as a green public object. It must look like a surface or a volume which is green; and so must have the visual appearance of a spatial thing. For a private object to have the visual appearance of a spatial thing entails it occupying a spatial region of one's visual field. So again, even if one allows private objects, the negation of ‘everything green has some spatial property’ entails a contradiction.

What goes for these examples goes, I suggest, for the general reason stated earlier and for all similar examples. So I stick to my view that it is their relation to contradictions which makes sentences logically impossible or necessary. Hence, they have no truth-conditions or no falsity-conditions, as the case may be; and they are such that they or their negations are inconceivable in the sense analysed above.

What may lie behind the objections of LeftowFootnote ¹⁴ and others to my view is the view that all valid proofs must accord with the rule of some recognized system of deductive logic, or consist of a demonstration that there is no model of the proof in which the premises are true and the conclusion is false. But it is an absurd requirement to require a proof to accord with some recognized system of logic. It follows from no logical system that ‘x is red’ entails ‘x is coloured’; and that is a lot more obvious than most entailments. Anyway, such a requirement gets things the wrong way around. Our grounds for believing that some system of logic yields only valid inferences is that it captures as entailments many entailments which we already recognize as such, and none (or almost none) which are clearly not entailments. (Since ‘almost none’ is vague, it requires a decision to determine that there are so few exceptions that we are justified in calling a system of logic a valid system. Apparent exceptions can then be dealt with by holding that the grammatical form of the sentences involved is misleading as a guide to their ‘logical form’.) Our recognition of so many entailments is prior to our recognition of some system of logic as capturing them.

It is equally absurd to require a demonstration of entailment by means of models. For a model is an interpretation of each of the ‘non-logical’ terms of premises and conclusion; and a demonstration of the validity of the argument consists in showing that there is no model in which the premises are true and the conclusion false. But the validity of a proof so often depends on the ‘non-logical’ terms involved – no model will show that ‘x is red’ entails ‘x is coloured’. (And anyway there is no clear understanding of what counts as a ‘non-logical’ term.)

Finding for some sentence s a sequence of mini-entailments that shows that that sentence entails a contradiction is, however, often a very difficult matter; and any particular route of proof may be challenged by someone disputing some purported mini-entailment from some p to q along the sequence. An opponent will challenge a claim that p mini-entails q by claiming that q is not something to which anyone is committed when using p in the correct sense. This disagreement may be overcome if the proponent can get his opponent to recognize some r such that p mini-entails r and r mini-entails q. Or the disagreement may be bypassed if the proponent can find a different sequence of mini-entailments from s to a contradiction which an opponent will recognize as such.

There is a way of attempting to prove that a sentence s is not logically necessary or (as the case may be) not logically impossible (other than by proving that it is impossible, or as the case may be, necessary), which involves an extra step. This involves showing that s or (as the case may be) not-s is logically possible (i.e. does not entail a contradiction). For if not-s is logically possible, then s is not logically necessary; and if s is logically possible, then not-s is not logically necessary. The extra step involves finding a sentence r which disputants agree to be logically possible. The argument then consists in showing that r entails s (or, as the case may be, not-s) by a sequence of agreed mini-entailments. For if r does not entail a contradiction, neither does any sentence entailed by it. So if disputants can agree that r is a logically possible sentence and that r entails s, they can agree that s is not logically impossible.

Philosophers often try to show that some supposition – for example ‘there are two spaces’, a space being a system of places each of which is at some distance in some direction from each other place of the system and from no other places – is logically possible, by describing in detail a situation under which it would be true. That is, they claim that the latter description entails that there are two spaces; and that since the latter description is logically possible, ‘there are two spaces’ is also logically possible. But the same problems about recognising entailments may arise, as before; and there is the additional need to secure agreement that r, which will typically be quite a long conjunction, is logically possible. But again if disputants cannot agree that some suggested starting point is logically possible or that certain mini-entailments hold, they may be able to find a different starting point with a different route of proof on which they can agree.

Yet long failure to agree on a sequence of mini-entailments and (if required) a logically possible starting point which shows a sentence s to be or not to be logically necessary or impossible (as the case may be), may show that the disputants are using words in different senses. One or other disputant may be using words in ways importantly different from the correct way, or there may be no one correct way. But when each of the two disputants is prepared to recognize that the other disputant is using words in a legitimate way, that is that their use has been introduced by paradigm examples and inferential rules such that some sentences containing those words used in the different sense have ultimate sense (i.e. do not entail contradictions arising from the senses of those words), then there is no substantial dispute between them.

For example, some writers have been introduced to the phase ‘epistemically justified’ by (among other paradigm examples) examples such as that a clairvoyant whose beliefs were produced by a reliable process is epistemically justified in holding them, and that if we had the same set of beliefs about the external world in an evil-demon world as we do now we would not be justified in holding them. These examples lead to a reliabilist epistemology in which a belief being ‘justified’ entails that it is produced by a reliable process. But other writers have been introduced to this phrase by many paradigm examples of people holding beliefs, including false beliefs, which in some a priori sense are made probable by the evidence accessible to them – which leads to an internalist epistemology of epistemic justification in which the justification of a belief depends solely on introspectible data, and does not entail that the belief is produced by a reliable process. But epistemologists are beginning to recognize that both senses of ‘epistemic justification’ are legitimate senses.Footnote ¹⁵

However, the dispute is more serious when one or other disputant is not prepared to recognize the other's usage as legitimate. The disputants could have been introduced to a word by somewhat different paradigm examples and mini-entailments. But they could have been introduced to the word by exactly the same paradigm examples and exactly the same set of mini-entailments, but extrapolate from them differently so as to produce the disputed mini-entailments. The longer the dispute lasts and proves insoluble by the attempts of one disputant to get the other to recognize the mini-entailments or initial starting points which he acknowledges, or to agree that they are each using words in different legitimate senses, the more probable it is that one or other disputant is suffering from an intellectual deficiency. This deficiency will be either that one disputant cannot recognize a further property manifest in the paradigm examples of the application of the word and its agreed mini-entailments, not logically equivalent to the one which both of them can recognize, or that one disputant is deceiving himself when he claims to recognize such a property.

Thus, most of us have been taught the meaning of the word ‘person’ by examples of the same kind (‘John is a person’, ‘You are a person’, ‘This is the same person as the person you saw last week’) and similar mini-entailments; (‘John is a person’ mini-entails ‘John is sometimes conscious’, and is mini-entailed by ‘John is human’). But we may recognize more than one similar property in the common paradigm examples, consistent with the paradigm mini-entailments. We will all recognize the paradigm examples as examples of space-occupying and sometimes conscious animate beings with a certain sophistication of thought. We will all also recognize the paradigm examples as examples of sometimes conscious animate beings with a certain sophistication of thought. We can all agree that ‘person’ can be defined either as a being of the former kind or as a being of the latter kind.

But some of us will think that someone being a person on either definition mini-entails their being a person on the other definition; while others of us will think that the two definitions pick out different properties, possession of the former mini-entailing possession of the latter but not conversely, and then prefer to define being a ‘person’ as being a being of the latter kind. Consequently, the first group will hold that ‘This person will exist tomorrow without occupying space’ (i.e. in a disembodied state) is logically impossible, whereas the second group will allow this as a logical possibility.

The first group will hold that the second group in claiming to have a sense of ‘person’ not logically equivalent to the former sense do not have a legitimate sense of ‘person’, and are deceiving themselves in thinking that they do. This – the first group will claim – is because every sentence containing the word ‘person’ used in the second sense entails a contradiction; e.g. ‘John is a person’ entails ‘it is impossible that John exists and does not occupy space, and it is possible that John exists and does not occupy space’. The second group, while acknowledging that the first group have a legitimate sense of ‘person’ (though not their preferred sense) claim that their latter sense is also a legitimate sense; they can see a property common to the paradigm examples exemplifiable without the feature of space-occupying. This disagreement will be difficult to resolve in the ways which I have been describing earlier of finding common starting-points or sequences of mini-entailments, and it may prove irresoluble.

A strong reason, however, why we can suppose that normally these methods will reach clear and agreed results is because we are investigating the meaning (as given by observable truth-conditions and mini-entailments) of sentences of a human language, to the rules of which we have been introduced by very similar training. We are very similar to each other in our psychological make-up; and so we would expect normally to agree with each other about the mini-entailments of sentences and so about which sentences are logically necessary (or impossible). It is only when some of us are blind to certain differences between properties or self-deceived, that this won't happen. And while for any sentence whatever seems to be true is – on that evidence – to some extent probably so (which is the claim of the principle of credulity), we have in this case a further consideration that increases the probability that most of our judgements about mini-entailments and so about logical necessity or impossibility are true. This consideration is that our judgements concern features of our own human language, and we know our way about our own language.

Evidence that we are well equipped and placed to make correct judgements about some matter increases the probability that the judgements we make are true. This fact about the nature of our judgements about these matters reinforces the probability that in general (when there is not significant disagreement) what seems logically possible, very probably is logically possible; and similarly for the necessary and impossible.

My conclusion from this long section is that logical necessity (and impossibility) and our access to it can be explained fully by the linguistic conventions governing sentences. Of course that is not to deny that God knows all necessary truths, in the sense that He knows at all times that if humans were to speak a certain language which sentences of that language would be necessarily true; nor is it to deny that God is the source of the ability of humans to speak whatever languages they speak.

The transcendent world

But many philosophers, among them Brian Leftow, believe that there is a much deeper realm of reality than the mundane realm of human sentences which I have described, on which necessary truth ultimately depends. This view acknowledges that humans may not always understand the truth-conditions of sentences in the same way as each other and may disagree about whether some sentence entails another one. But, fortunately, on this view there are languageless things which constitute the truth- and falsity-conditions of necessary or impossible human sentences.Footnote ¹⁶ I will discuss this view initially in terms of the variant that such sentences express eternally true or false ‘propositions’ which exist in a Platonic heaven; then I will apply my results briefly to the variant that these propositions are thoughts in the mind of a necessarily existing God,Footnote ¹⁷; and finally come to Leftow's own variant of what is the locus of the truth- and falsity-conditions of necessary and impossible sentences.

Before going further, I need to draw attention to different languageless entities which may be described as ‘propositions’. LemmonFootnote ¹⁸ distinguishes between what he calls a ‘proposition’ (which in order to avoid confusion with a wider sense of ‘proposition’ I shall call ‘L-proposition’), and what he calls a ‘statement’. The L-proposition which a token sentence s expresses is that element of claim in what is said which would also be made by any other token sentence which is synonymous with s. Two sentences express the same L-proposition if and only if they are synonymous. ‘Rex mortuus est’ uttered by a Latin speaker of the fourteenth century, ‘Le roi est mort’ uttered by a French speaker of the eighteenth century, and ‘The king is dead’ uttered by an English speaker of the twentieth century would then (on normal understandings of their meanings) express the same L-proposition. An L-proposition plus its context of utterance determines which statement is made. A statement (as Lemmon understands it) is that element of claim in what is said which concerns the actual individuals, properties, times, places etc. picked out by the referring expressions, independently of how they are picked out. The three token sentences just cited did not express the same statement, for (we may reasonably suppose) they concern different kings. Whereas ‘I am ill’ spoken of me, ‘You are ill’ addressed to me, and ‘He is ill’ spoken of me, do all express the same statement (although not the same L-proposition.). An L-proposition and a statement are both to be distinguished from the ‘thought’ (in what I take to be Frege's sense) which a token sentence s expresses, which is that element of claim made by any other token sentence which expresses both the same L-proposition and the same statement as s.

Those token sentences whose truth-value does not depend on any non-linguistic conditions, including those which determine the reference of their referring expressions, express logically necessary or logically impossible L-propositions and thoughts. Those sentences which, given the actual objects or properties picked out by their referring expressions (e.g. given the planet actually picked out by ‘Hesperus’), have a truth-value independent of all other non-linguistic conditions express a posteriori metaphysically necessary or impossible statements and thoughts. Thus, given (and only given) the actual reference of the referring expressions in the sentence ‘Hesperus is Phosphorus’ (which is that both ‘Hesperus’ and Phosphorus' refer to a planet formed of the same chunk of matter), that sentence expresses a necessary statement and a necessary thought.

Now certainly it is useful in various philosophical contexts to talk as if there were such entities as these. For example, if we want to talk about a claim made by some sentence, and we want what we say to apply to the claim made by any other sentence which attributes the same properties to the same objects at the same time, it is more convenient to talk about the claim made by the statement expressed. But the Platonist claims that L-propositions and/or statements and/or thoughts really exist in a Platonic heaven, marked with their logical status (e.g. ‘logically necessary’) whether or not any human language does or could express them. Having made the distinctions between L-propositions, ‘statements’ etc. in order to bring out the different languageless things which might be supposed to be expressed by sentences, I shall in future ignore them and make my points merely about ‘propositions’, understood as covering all or any of these.

Yet, as I have illustrated, we can analyse metaphysical necessity and impossibility in terms of token sentences, the circumstances under which speakers would regard them as true (false), and the commitments to other sentences which speakers are considered to have when they utter those sentences. Why postulate more? One answer sometimes given is that propositions would explain why speakers behave as they do. Speakers have some grasp of this eternal web of propositions; they seek to express propositions in language and to conform in their human inferences with human sentences to the entailments of eternal propositions. But there's an easy explanation of human linguistic behaviour which involves no such apparatus. Humans want to know what the world is like for theoretical and practical reasons; and to communicate information about this to others and receive it from others. For this purpose they need sentences with fairly clear content.

The content of a sentence will only be clear to the extent to which the sentence has sharp truth- and falsity-conditions. Hence the need for agreement about paradigm examples and mini-entailments. The existence or nonexistence of propositions wouldn't make this behaviour any more or less comprehensible. Further, if the necessity of propositions depended on some transcendent non-linguistic fact, there would be no additional good reason (other than that provided by the principle of credulity) to suppose that we would be able to recognize that necessity. Our access to how human language works is better than our access to the transcendent, and so our ‘intuitions’ about the former are more to be trusted than our intuitions about the latter. To suppose that there are propositions would involve supposing that we know less about necessary truths than surely we do. All told, there are no grounds for postulating the existence of propositions as Platonic entities.

But theism opens the way to variants of a transcendent world view which make necessary truths and impossibilities, not independent entities, but ones grounded in the activity or nature of a God to which theism is already committed. One variant is that propositions are God's thoughts, necessary propositions being the ones which he (in some sense) necessarily thinks and whose necessity depends on God thinking them. Brian Leftow's variant avoids postulating propositions by holding that the truth-makers of necessary truths are ‘ultimately divine mental events and powers’, divine events being ‘understood in a broadly functionalist’ way.Footnote ¹⁹ Then what makes a token sentence ‘No squares are round’ necessarily true is that God's nature makes it the case that whoever utters such a sentence utters a true sentence.

A theist holds that all existent things and their properties depend on God as the necessary condition (and except insofar as the free will of creatures is also involved, the sufficient condition) of their existence; and so if a theist also holds that necessary truths have transcendent truth-conditions, he will want to locate these truth-conditions in God in some such way. But it is a natural reaction to any such theory to say that surely ‘No squares are round’ is true whether or not God thinks it or has the appropriate mental state. And if the truth of ‘No squares are round’ is just the truth of a human sentence whose truth results from human conventions, there is no need for a theist to deny that. For, given that there is a God, it will be because of God's creative act that humans exist and establish linguistic conventions and so utter sentences which are necessarily true (or false). It remains the case that there is no reason to postulate transcendent truth-conditions for necessary and impossible truths.

God's necessity

Quite a number of religious thinkers since Anselm have thought of God as a logically necessary being, and in his paper Leftow gives a tentative argument in favour of this view (in his terminology a tentative argument to show that God's existence is ‘narrow-logically necessary’ (141).) Given the results of the first section of this paper, God would be a logically necessary being if and only if the sentence ‘There is no God’ entails a contradiction. Those who put forward an ontological argument have attempted to prove that there is a God by claiming that ‘There is no God’ does entail a contradiction. But notoriously, ontological arguments are either invalid or have a highly dubious premise (e.g. Leftow's premise (15) ‘Necessarily if God exists, He exists necessarily’ (158)). Another example of a similarily dubious premise is the premise in the ontological argument put forward by Plantinga, ‘There is a possible world in which maximal greatness is instantiated’. Plantinga defines a being as having ‘the maximal degree of greatness in a given world W only if it has maximal excellence in every possible world. Maximal excellence entails omnipotence, omniscience, and moral perfection.’Footnote ²⁰ The apparently modest premise then yields the strong conclusion that there actually exists a maximally great being.

An advocate of an ontological argument may attempt to show that the premise deemed dubious by others is a necessary truth by deriving a contradiction from its negation by means of mini-entailments, but few doubters are likely to be convinced by the attempt. But there is a very powerful argument for the logical possibility of ‘There is no God’ from the strong a priori obviousness that some descriptions of a godless world do not entail a contradiction. Surely there is no contradiction entailed by the existence of a world in which there are just four atoms not caused to begin or to continue to exist by anything or anyone else. Spelling out what it would be like for this suggested world or some other similar uncaused physical world to exist will surely in the course of time get almost anyone to admit that there is no contradiction entailed by the existence of one such world. The existence of such a world entails the nonexistence of God, the supposed cause of the existence of all substances. It follows from the logical possibility of such a world that ‘There is no God’ is logically possible and so (given S5) that it is logically necessary that it is not logically necessary that there is a God. So not merely is the dubious premise of a typical ontological argument false, but necessarily if there is a God, His existence is logically contingent.

But why should not ‘There is a God’, read as ‘God exists’ where ‘God’ is a rigid designator of an individual whose essential properties we do not know, be a posteriori metaphysically necessary, in Leftow's terminology a posteriori ‘broadly logically necessary’? Leftow claims that this is so, as have many others in recent years. Given my claim at the beginning of the paper that a sentence is a posteriori metaphysically necessary iff it reduces to an a priori necessity by substitution of canonical descriptions of some essential properties of the properties or objects rigidly designated, it follows that ‘God exists’ would be a posteriori metaphysically necessary iff it reduces to an a priori necessity when we substitute for ‘God’ some canonical descriptions of His essential properties.

Even if we did not know any of the essential properties of God, any theist must hold that (if there is a God) the actual properties of God include being an all-powerful and all-good cause of the world. Hence the existence of a world with four uncaused atoms (or any similar supposition) entails the nonexistence of any being with any essential properties at all (however described) who has the actual properties of being an all-powerful and all-good creator. And so the logical possibility of the four-uncaused-atoms world (or any similar world) entails the logical impossibility of a (either a posteriori or a priori) metaphysically necessary God. So I do not need such a precise thesis of the necessary and sufficient conditions of a posteriori necessity as that stated at the beginning of this paper in order to get this result.

But if there were to be logically necessary eternal propositions, and especially ones whose necessity does not arise from their negation entailing a contradiction, or sentences which have their truth-conditions in aspects of the divine nature, then it seems prima facie coherent to suppose that some sentence somewhat similar to Leftow's premise (15) might be or expresses one of these. Maybe there is a proposition which (15) expresses which is logically necessary (or a divine mental event which makes (15) logically necessary) without the proposition or sentence entailing a contradiction. And if one thinks that there could be propositions or sentences which are a posteriori metaphysically necessary for reasons other than that those sentences or the sentences which express the propositions express a priori necessities when canonical descriptions of some essential properties are substituted for rigid designators, maybe ‘God exists’ could be or express an a posteriori necessary truth. It's one person's intuition against another person's intuition; further argument does not seem possible.

We shall, however, only be in this situation of a clash of irresoluble intuitions, if we start postulating strange metaphysical entities or find truth-conditions in apparently irrelevant places for no good reason. If it is only human conventions which make sentences necessarily true or false, it is not likely that we shall be very often in the situation of a clash of irresoluble intuitions about what are the metaphysically necessary truths. We may, however, as we saw earlier, occasionally be in this situation because some people extrapolate from the teaching process in ways which other people cannot understand, or mistakenly think that they are extrapolating in a way which yields a legitimate sense. But because we all have a similar psychological make-up and belong to a language-community, that will not happen often. And for reasons given earlier, it is plausible to suppose that almost everyone can eventually come by the method described earlier to see that ‘There is a God’ is not a logically necessary sentence. That is not to deny that God is a necessary being in the sense which all theists would endorse that His existence is not causally contingent on anything else (i.e. nothing else causes Him to exist).